Your change of coordinates turned the ellipsoid into a sphere, but it messed your distance function: Instead of having spheres as surfaces of equidistance you now have ellipsoids, so you didn't make the problem any easier.

You can probably try to minimize the square of the distance, subject to the solution being on the ellipsoid. This can be done using a Lagrange multiplier, and everything should be quadratic in the equations involved, so it should be easy to solve.

The shortest distance is ever along a normal on the ellipsoid surface. See e.g. David's short paper.

(disregard.)

The shortest distance is ever along a normal on the ellipsoid surface. See e.g. David's short paper.

Yes, that PDF is quite short. Your post inspired me to make it longer . The similar point-ellipse PDF already had a lot of the ideas for computing the distance robustly. I originally mentioned the inspiration for the PDFs and my source code was a Graphics Gems IV article. That article converts the problem to computing the largest root of a polynomial (degree 4 for the ellipse, degree 6 for the ellipsoid). The suggestion is to us Newton's Method with an initial guess greater than the largest root, and the authors provided an initial guess. Unfortunately, the convergence of the method is not guaranteed. In particular, the problems occur when the query point is near a coordinate axis (for the ellipse) or near a coordinate plane (for the ellipsoid). This showed up in my point-ellipse code, which I had then rewrote to include special handling for such points. I never got around to doing the same thing for the point-ellipsoid code.

A new PDF combines the point-ellipse and point-ellipsoid PDFs, but has a greatly expanded discussion on robustness and on why you do not want to use the polynomial idea. The numerical problems occur as a result of a topological change in the graph of a function F(t) (I defined this in the PDFs). In terms of the polynomial P(t), the problem is that points near the coordinate axes/planes cause P(t) to have a largest root that is multiplicity 2 when you are on the axes/planes. Newton's Method needs to be modified to handle roots of multiplicity larger than 1, but even so, finding the largest root of F(t) is much easier to make robust.

The new PDF is Distance from a Point to an Ellipse, an Ellipsoid, or a Hyperellipsoid . I updated my WM5 source code to use this and I added a new sample application to illustrate. The application has the code for point-hyperellipsoid (any dimension you want).